On the symmetry classification of integrable evolution equations of the 3rd order

Abstract

• We present new results in the framework of symmetry classification of integrable evolution vector equations of the 3rd order. A technique proposed by G.A. Meshkov and V.V. Sokolov allowed us to find 12 equations satisfying the necessary integrability conditions. We provide a short review of all known nowadays equations of the considered type and also clarify all computational difficulties not allowing us to complete the classification problem in the general form. By imposing reasonable additional restrictions for the form of equations while classifying them we succeed to complete the calculations. The found equations possess several nontrivial preserved densities and they are likely exactly integrable. As the proof of their integrability, the Lax representation or Backl\"und autotransform could serve but to find them is a rather complicated problem requiring a sufficient motivation, for instance, an application value of some of these equations.